Tham khảo tài liệu 'diffusion solids fundamentals diffusion controlled solid state episode 1 part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Random Walk and Diffusion 61 n 1 n R2 E r 2E 12 r-j i 1 i 1 j i 1 n 1 n X2 Xi2 2E 12 - XZ i 1 i 1 j i 1 The first term contains squares of the individual jump lengths only. The double sum contains averages between jump i and all subsequent jumps j. Uncorrelated random walk Let us consider for the moment a random walker that executes a sequence of jumps in which each individual jump is independent of all prior jumps. Thereby we deny the walker any memory. Such a jump sequence is sometimes denoted as a Markov sequence memory free walk or as an uncorrelated random walk. The double sum in Eq. contains n n 1 2 average values of the products xXj or riTj . These terms contain memory effects also denoted as correlation effects. For a Markov sequence these average values are zero as for every pair XjXj one can find for another particle of the ensemble a pair xiXj equal and opposite in sign. Thus we get from Eq. for a random walk without correlation RLdom 12 tói i 1 random Ẻ tá i 1 The index random is used to indicate that a true random walk is considered with no correlation between jumps. In a crystal lattice the jump vectors can only take a few definite values. For example in a coordination lattice coordination number Z in which nearest-neighbour jumps occur jump length d with x-projection dx Eq. reduces to RLdom d1 Random dX Here n denotes the average number of jumps of a particle. It is useful to introduce the jump rate r of an atom into one of its Z neighbouring sites via r We then get D 1 d2zr 6 6t 62 4 Random Walk Theory and Atomic Jump Process Table . Geometrical properties of cubic Bravais lattices with lattice parameter a Lattice Coordination number Z Jump length d Primitive cubic 6 a Body-centered cubic bcc 8 aZ3 2 Face-centered cubic fcc 12 aZ2 2 This equation describes diffusion of interstitial atoms in a dilute interstitial solid solution1. The quantity T A i is the mean residence time of an atom on a certain
